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Chasing the Elusive Pearson Distribution: Skewness and
Kurtosis in Financial Markets
Artemis Econometrics, LLC
www.artemis-econometrics.com
[email protected]
Draft submitted for comments July 8, 2013.
© 2013 Artemis Econometrics, LLC. All rights reserved.
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Contents
Introduction ..................................................................................................................................... 3
The Normal Distribution ................................................................................................................. 3
The Pearson Type IV Distribution .................................................................................................. 5
Probability Density Function .......................................................................................................... 6
Parameter Estimation ...................................................................................................................... 7
Recent Research .............................................................................................................................. 8
Application to Daily Returns .......................................................................................................... 9
Statistical Significance of Pearson Parameters ............................................................................. 11
Application to Regression Analysis .............................................................................................. 12
Comparison to Robust Regression ................................................................................................ 14
Conclusion .................................................................................................................................... 16
Tables ............................................................................................................................................ 17
Figures........................................................................................................................................... 19
References ..................................................................................................................................... 22
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Introduction
For at least half a century, there has been compelling evidence that asset returns are not
normally distributed but rather are subject to tendencies in direction (skewness) and extremal
events (kurtosis). Despite the general awareness that market returns do not follow a normal
distribution, integrating skewness and kurtosis into time series analysis remains a difficult task.
One intriguing possibility for combining skewness and kurtosis with traditional mean and
variance is the flexible, asymmetric, fat-tailed Pearson Type IV (henceforth PearsonIV)
distribution. Important mathematical and programming developments have made the PearsonIV
distribution accessible to a wider audience, and researchers have recently applied several
PearsonIV methodologies to financial markets.
This study describes the PearsonIV distribution, examines its out-of-sample predictive
properties, and uses it as a basis for regression analysis. The results are encouraging, although
proper use of the PearsonIV distribution requires a thorough understanding of the strengths and
potential pitfalls of its plasticity.
The Normal Distribution
The normal distribution, a continuous probability distribution defined by mean and
variance, forms the centerpiece for many tools in investment management. Most prominently,
the normal distribution’s standard deviation is the most common numerical representation of risk
in financial markets.
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Despite its widespread use, there is a general understanding that the normal distribution’s
mean and variance do not adequately explain
the dispersion of returns on financial markets.
The normal distribution is symmetric around
its mean, but asset returns often seem to occur
more frequently in one direction. The normal
distribution also has “thin tails,” meaning it
ascribes a relatively low probability to large
returns in either direction. The shortcomings
of the normal distribution are especially
apparent when extreme returns occur more
frequently than would be predicted by the
distribution’s thin tails.
There are conventional measures of
the asymmetry of returns (skewness) and the
relative probability of events outside the norm
(kurtosis). Both skewness and kurtosis have
been studied extensively. For instance,
Campbell, Lo and MacKinlay (1997) cite
several decades of overwhelming evidence
that financial returns exhibit high kurtosis.
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Integrating skewness and kurtosis with ordinary mean and variance has been challenging.
Campbell, Lo and MacKinlay recount a long history of attempts to apply non-normal probability
distributions to financial time series. Most of these efforts were eventually abandoned. Campbell,
Lo and MacKinlay end their narrative in the late 1990s, suggesting that the Student’s
distribution and mixtures of distributions are worthy of further study.
The Pearson Type IV Distribution
In 1895, Karl Pearson defined a collection of curves that became known as the Pearson
family of distributions. The fourth distribution has “unlimited range in both directions and
skewness,” and has recently received attention for its potential application to financial markets.
The PearsonIV distribution is defined by four real numbers ( ) that interact with each
other to offer a wide range of shapes and sizes.
The parameter is a location metric similar to the mean of a normal distribution. The
variable is a measure of scale, analogous to standard deviation. The value of specifies the
symmetry of the distribution. The parameter determines kurtosis, with smaller values of
creating thicker tails.
The range of possible shapes is so expansive that the PearsonIV actually encompasses
other distributions. When , the PearsonIV is symmetric and becomes a Student’s
distribution. When and , the PearsonIV approaches the normal distribution in the
limit. When , the PearsonIV becomes an asymmetric Cauchy distribution.
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Probability Density Function
The PearsonIV probability density function (PDF) is:
( ) [ (
)
]
[ (
)] ( ⁄ ) (1)
The PDF, and the likelihood function to be examined later, require a normalizing
constant that depends on the values of , and . The difficulty in calculating likely
impeded the use of the PearsonIV distribution for over a century. The orthodox form of the
PearsonIV normalizing constant works in complex numbers (i.e., numbers that have an imaginary
component defined as the square root of -1). Moreover, it requires a gamma function that can
manipulate a complex number, which most statistical packages do not provide.
To avoid the complex gamma function, Nagahara (1999) uses an infinite multiplication
series. However, this is probably too burdensome for routine analysis.
Particle physicist Joel Heinrich (2004) provides a major advancement, and nearly all
equations and descriptions of the PearsonIV distribution in this study come from Heinrich’s work.
Heinrich reformulates the most problematic part of the normalizing constant in terms of the
hypergeometric function and provides sample code to calculate . This study uses C++ code
modified from Heinrich’s example to compute . Heinrich also defines the PearsonIV cumulative
distribution in terms of the hypergeometric function and offers programming routines to generate
random numbers from the PearsonIV distribution.
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Parameter Estimation
To determine the PearsonIV parameters , , and , one first calculates the mean,
variance, skewness and kurtosis of the series . From these, denoted by , , and , the
PearsonIV estimates are:
( ) ( )
(2)
( )
√ ( ) ( ) (3)
√ [ ( ) ( ) ]
(4)
( ) √
(5)
Note that is used as an abbreviation for ( ).
It is usually preferable to estimate the parameters jointly by minimizing the negative log
likelihood:
∑ [ (
)
]
∑
(
) (6)
Transforming the PearsonIV parameters , , and into traditional mean, variance,
skewness and kurtosis requires that be at least 1.0, 1.5, 2.0 and 2.5, respectively. Depending
on the data set, these constraints on may preclude solutions that might otherwise be chosen.
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If an analysis only uses the PearsonIV distribution, must be greater than 0.5. This
minimal constraint should not be restrictive for financial data.
Recent Research
The PearsonIV distribution has attracted significant attention in recent years. Premaratne
and Bera (2001) suggest that compared to the Student’s distribution, the PearsonIV has heavier
tails and can account for skewness. They also advise that the PearsonIV “is much easier to handle”
than other asymmetric possibilities such as the non-central and the Gram-Charlier distributions.
Brännäs and Nordman (2003) add to an established record of tests rejecting the normal
distribution when applied to financial returns. They find that the one-parameter log-generalized
gamma distribution indicates skewness in financial data, while the “PearsonIV model suggests that
excess kurtosis rather than skewness should be accounted for.”
Yan (2005) shows that the PearsonIV distribution has a much larger range of skewness and
kurtosis combinations than the Gram-Charlier distribution, which has also been used to model
skewness and kurtosis simultaneously.
Grigoletto and Lisi (2007) and Bhattacharyya, Misra and Kodase (2009) apply the
PearsonIV distribution to value-at-risk analysis.
Grigoletto and Lisi (2009) propose a dynamic, time-varying PearsonIV model in which the
parameters are defined by nine GARCH coefficients. This approach is applicable to very long
time series and allows the shape of the PearsonIV distribution to change through time within a
data set.
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Stavroyiannis, Makris, Nikolaidis and Zarangas (2012) use the PearsonIV model for value-
at-risk analysis, and find that “the PearsonIV distribution gives better results, compared with the
skewed student [t] distribution, especially at the high confidence levels, providing a very good
candidate as an alternative distributional scheme.”
Application to Daily Returns
This study models the PearsonIV parameters directly from the data, without the
intervening GARCH filter used by Grigoletto and Lisi. The parameters are therefore constant
throughout the estimation time frame. This approach is robust even with small data sets and
reflects what many financial professionals do in practice (e.g., calculating standard deviation or
skewness from a relatively small sample of recent returns).
Figure 1 demonstrates this method in practice. The solid line represents the normal
distribution of an S&P 500 exchange traded fund (ETF) on October 31, 2008, fit to one year of
daily returns. This example takes place during the financial crisis, when the stock market
witnessed some of the largest volatility in its history. The standard deviation of daily returns over
the previous year was 2.22%, and its mean daily return was -0.14%.
The hashed line signifies the PearsonIV distribution fit to the same data.
The numbers along the axis do not indicate actual probabilities. Rather, the height of
these curves denotes the relative probability of observing a return at any point along the axis.
The axis has a high scaling because the axis, representing daily returns, has a small scaling,
and the area under each curve must sum to one.
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The solid line drawn upward from the horizontal axis corresponds to the 0.28% return of
the S&P 500 ETF on the next trading day, November 3, 2008. The and symbols reflect the
relative likelihood of this return given the probability densities derived from daily returns over
the prior year. The PearsonIV distribution allows for a much higher probability of the November 3
return than does the normal distribution. This dramatic example illustrates the potential costs of
applying a normal distribution to non-normal financial series.
The normal distribution expands outward in an attempt to account for the extreme
volatility of the S&P 500 index during this time. However, there are significant negative
consequences. Simply increasing the variance of a normal distribution does not change its nature
as a thin-tailed probability density. Therefore, the normal distribution is not able to put enough
weight in its tails to match the PearsonIV distribution’s kurtosis.
Moreover, the bulky middle section of the normal distribution stretches along the axis,
meaning it has a density that is too low around its center. It therefore attributes a relatively low
probability to events occurring around its mean, such as the November 3 return of 0.28%.
Financial returns tend to have frequent returns around zero and occasional extreme
returns. The PearsonIV distribution is able to form a shape that has a very dense section around
the mean but also thick tails. In contrast, significant density weight in the normal distribution is
shifted to intermediate areas, where the probability might actually be quite low. The normal
distribution would be more effective if there were more probability weight directly in the middle
and on the tails.
An exercise of this sort can indicate whether one distribution is consistently more
effective at accounting for the observed return pattern. Table 1 presents the results of this
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exercise repeated every trading day, data permitting, from January 2000 to March 2013. At each
point, the probability densities of the normal and PearsonIV distributions are calculated using
daily returns from the previous year. These densities are then compared to each other using the
next day’s actual return, and these results are summed through time. Table 1 indicates that in all
cases the PearsonIV distribution offered a relative advantage in fitting a collection of ETFs and
hedge fund indices.1 The results may seem marginal, but this is because most of the time there is
little difference between the normal and PearsonIV distributions. They tend to diverge
substantially during high market volatility, as demonstrated in Figure 1.
Statistical Significance of Pearson Parameters
Figure 2 shows rolling statistics of the PearsonIV evaluated for an S&P 500 ETF using
three years of weekly data. The statistics are based on variance estimates taken from the
negative inverse of the matrix of numerical partial second derivatives (Hamilton, 1994).
The results are consistent with Brännäs and Nordman (2003), who find that the kurtosis
parameter is more significant than the skewness parameter when the PearsonIV distribution is
applied to equity returns.
Not only is the symmetry parameter the least significant statistically, the sign remains
positive even during the financial crisis of 2008. The PearsonIV parameters are estimated jointly,
so symmetry is defined relative to the location parameter . It is possible for the PearsonIV to
have a negative mean, but a positive skew, during a market sell-off. This means that even though
returns are mostly negative, when they are positive they are larger in the positive direction. This
1 Source: Hedge Fund Research, Inc., www.hedgefundresearch.com, © 2013 Hedge Fund Research, Inc. All rights
reserved.
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is what happened during this time, for the two largest daily percentage gains in the history of the
S&P 500 index were 9.93% and 10.79% increases on October 13 and October 31, 2008,
respectively (Wikipedia: List of largest daily changes in the S&P 500).
Although the concept of skewness has a compelling appeal, the statistical results can be
ambiguous. Skewness tends to be less statistically significant than kurtosis, and the interpretation
of skewness can be counterintuitive when it is defined relative to a mean, rather than with respect
to zero.
Application to Regression Analysis
Regression analysis presents a potentially useful application of the PearsonIV distribution.
The concept is similar to robust regressions, which have long been used to minimize the effects
of outlier data points on the regression coefficients. Allowing regression residuals to have a
PearsonIV distribution would not only account for outliers, it would also help to keep systematic
distortions in the shape of the residuals’ distribution from affecting the coefficients.
Ordinary least squares (OLS) regression takes the form:
( ) (7)
A PearsonIV regression is defined as:
( ) (8)
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The PearsonIV residuals are allowed to have an off-center mean with a non-normal shape.
Figure 3 shows rolling OLS and PearsonIV beta estimates using the Hedge Fund Research HFRX
Global Hedge Fund Index2 as the dependent ( ) variable, and an S&P 500 ETF as the
independent ( ) variable. Both regressions use two-year rolling windows of weekly data and
include one lag term.
The most striking result is how quickly the PearsonIV model was able to react to the
decreasing equity exposure of this hedge fund index after the 2008 financial crisis.3 The OLS
beta does not fully reflect changes in the portfolio until observations at the beginning of the
financial crisis drop out of the data set.
The residuals show why the PearsonIV regressions are much quicker to respond to the
changing market exposure of the hedge funds. Figure 4 displays the distributions of the residuals
from the January 29, 2010, regressions. The OLS residuals are normally distributed with zero
mean, while the PearsonIV residuals are allowed a non-zero mean and non-normal shape.
The difference in the coefficients results from the heavy weight the PearsonIV distribution
puts into the negative tail. During the financial crisis, both the S&P 500 and the hedge fund
index had negative returns. In an OLS regression, those two sets of negative returns are aligned
with each other and attributed to the coefficients as long as the observations are in the sample.
In contrast, the PearsonIV regressions can take a more flexible approach. As more
observations appear indicating the hedge funds have decreased their S&P 500 beta, the PearsonIV
2 Source: Hedge Fund Research, Inc., www.hedgefundresearch.com, © 2013 Hedge Fund Research, Inc.
All rights reserved. 3 This Hedge Fund Research, Inc., index is asset weighted based on the total assets of each strategy in the
hedge fund universe. Therefore, two explanations for the decreasing S&P 500 beta are that investors
shifted to lower-volatility strategies, or the underlying hedge funds reduced equity holdings.
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approach allows itself to unlink the negative returns of the index and the S&P 500 ETF that
occurred at the beginning of the financial crisis. The negative returns of the hedge fund index
then become negative residuals and show up as weight on the left-hand side of the residuals’
distribution.
This result is notable because several complicated methods have been proposed to model
hedge fund returns in a manner that responds quicker to changes in the underlying portfolio than
rolling OLS regressions. Here, the PearsonIV model is able to accomplish this in a very natural
way, even in the context of rolling regressions.
One other major difference between the OLS and PearsonIV regressions in Figure 3 is that
the PearsonIV model results in lower S&P 500 betas during the steadily rising market up to mid-
2008. This hedge fund index includes a broad range of investment styles, including low-volatility
absolute return strategies that should have little discernible equity exposure. The OLS beta above
0.50 could be too high for a diversified index of this sort. Hedge funds were posting positive
returns during this time because numerous asset classes were rising, not just equities. It is
possible that the OLS coefficient reflects the correlations of multiple well-performing strategies,
while the PearsonIV beta may represent a more realistic assessment of direct equity exposure in
the hedge fund index.
Comparison to Robust Regression
The conceptual difference between the OLS and PearsonIV regressions is similar to the
difference between OLS and robust regressions. Robust regressions iteratively reweight the
observations to generate coefficient estimates that are not heavily influenced by outlier data
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points. The goal of robust regressions is to produce coefficients that reflect the common
relationship between variables since they are not pulled in the direction of atypical or extreme
events. The PearsonIV regression model theoretically could account not only for unusual data
points but also for abnormal shapes in the distribution of the data.
Figure 5 presents a comparison of the OLS, PearsonIV and robust regression methods. The
Russell 2000 Value ETF return series contains several observations at the beginning of the
financial crisis that distort its true beta to the S&P 500. The OLS regression is unable to recover
from these outliers, and the OLS beta is too low as long as those data points remain in the
sample.
The robust regression handles the outliers more effectively than OLS, and gives a beta
estimate closer to its true value. However, it does not provide as much improvement as the
PearsonIV method. Furthermore, the robust approach produces erratic results over other parts of
the sample, while the PearsonIV coefficients remain stable.
This exercise uses the Huber robust beta that is the first option in James P. LeSage’s
MATLAB econometrics toolbox. There are several types of robust regressions, and they have
different tolerance settings. This example is not comprehensive enough to draw broad
conclusions regarding the efficacy of the PearsonIV approach versus robust regressions. However,
the results are encouraging, because they indicate the PearsonIV regression model might be an
attractive option to handle data that contain outliers or have odd distributions.
Table 2 presents the results of this exercise applied to a broad array of hedge fund
indices. The study takes a hedging perspective: lower mean absolute deviations and root mean
squared errors are better, while a higher out-of-sample R-squared is better.
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Conclusion
Convincing evidence over an extended time shows that the mean and standard deviation
of the normal distribution do not properly describe the dispersion of asset returns, especially
equity returns. The PearsonIV distribution offers an intriguing possibility for integrating mean,
variance, skewness and kurtosis in a coherent framework that allows a great deal of flexibility in
modeling asset returns.
This study provides evidence that the PearsonIV distribution can be more effective than the
normal distribution at modeling the unconditional returns of a range of ETFs and hedge fund
indices.
Furthermore, regressions with PearsonIV residuals may, under certain circumstances, offer
more accurate and stable coefficient estimates than OLS or robust regressions.
Early indications from this and other works show the PearsonIV distribution has both the
flexibility and robustness to compensate for limitations of the normal distribution. If future
research continues to demonstrate the value of the PearsonIV distribution, it could someday
become an important addition to the toolkit of academic researchers and financial professionals.
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Tables
Table 1
Relative Fit of Pearson Type IV and Normal Distributions
Exchange Traded Fund Ticker
Study Size
(Daily Data)
Relative Fit
(Pearson Type IV /
Normal)*
SPDR S&P 500 SPY 3329 1.072
PowerShares QQQ QQQ 3329 1.055
iShares Dow Jones US Real Estate IYR 3212 1.067
Financial Select Sector SPDR XLF 3329 1.060
Energy Select Sector SPDR XLE 3329 1.041
iShares S&P Global 100 Index IOO 3091 1.073
iShares Russell 3000 Growth Index IWZ 3184 1.065
iShares Russell 3000 Value Index IWW 3179 1.068
iShares Russell 1000 Growth Index IWF 3227 1.066
iShares Russell 1000 Value Index IWD 3227 1.064
iShares Russell 2000 Growth Index IWO 3184 1.030
iShares Russell 2000 Value Index IWN 3184 1.032
iShares S&P SmallCap 600 Growth IJT 3184 1.031
iShares S&P SmallCap 600 Value Index IJS 3184 1.027
iShares S&P Europe 350 Index IEV 3184 1.059
iShares Dow Jones Select Dividend Index DVY 2361 1.070
iShares MSCI EAFE Index EFA 2912 1.057
iShares MSCI Emerging Markets Index EEM 2505 1.069
HFRX Hedge Fund Index**
Identifier
Study Size
(Weekly Data)
Relative Fit
(Pearson Type IV /
Normal)
Convertible Arbitrage HFRXCA 521 1.075
Relative Value Arbitrage HFRXRVA 521 1.059
Market Directional HFRXMD 456 1.151
Macro/CTA HFRXM 521 1.048
Global Hedge Funds HFRXGL 521 1.145
Event Driven HFRXED 521 1.072
Equity Hedge HFRXEH 521 1.104
Equal Weighted Strategies HFRXEW 521 1.135
Equity Market Neutral HFRXEMN 521 1.033
Merger Arbitrage HFRXMA 521 1.190
Absolute Return HFRXAR 456 1.035
* Values greater than one indicate relative advantage to Pearson Type IV.
** Source: Hedge Fund Research, Inc., www.hedgefundresearch.com,
© 2013 Hedge Fund Research, Inc. All rights reserved.
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Table 2
Regression Performance: HRFX Hedge Fund Indices* Hedged with S&P 500 ETF
Mean Absolute Deviation (weekly %) OLS Pearson IV Robust Regression
Convertible Arbitrage 0.83 0.61 0.64
Relative Value Arbitrage 0.51 0.48 0.48
Market Directional 0.70 0.65 0.65
Macro/CTA 0.85 0.78 0.80
Global Hedge Funds 0.43 0.40 0.41
Event Driven 0.45 0.42 0.45
Equity Hedge 0.55 0.53 0.54
Equal Weighted Strategies 0.38 0.34 0.35
Equity Market Neutral 0.48 0.47 0.47
Merger Arbitrage 0.36 0.34 0.35
Absolute Return 0.31 0.31 0.32
Root Mean Squared Error (weekly %) OLS Pearson IV Robust Regression
Convertible Arbitrage 1.66 1.29 1.33
Relative Value Arbitrage 0.88 0.80 0.81
Market Directional 1.04 0.96 0.97
Macro/CTA 1.21 1.11 1.13
Global Hedge Funds 0.67 0.62 0.65
Event Driven 0.68 0.63 0.69
Equity Hedge 0.80 0.76 0.81
Equal Weighted Strategies 0.60 0.52 0.56
Equity Market Neutral 0.68 0.68 0.68
Merger Arbitrage 0.58 0.59 0.59
Absolute Return 0.44 0.46 0.48
Out-of-Sample R-Squared (%) OLS Pearson IV Robust Regression
Convertible Arbitrage 8.57 16.51 9.84
Relative Value Arbitrage 21.90 22.53 21.74
Market Directional 49.77 48.79 47.99
Macro/CTA 1.74 3.23 2.10
Global Hedge Funds 43.88 42.33 41.13
Event Driven 52.93 53.68 52.50
Equity Hedge 60.68 60.71 59.03
Equal Weighted Strategies 34.04 35.22 32.52
Equity Market Neutral 1.76 1.39 2.02
Merger Arbitrage 37.96 34.70 35.02
Absolute Return 6.21 4.71 5.72
* Source: Hedge Fund Research, Inc., www.hedgefundresearch.com,
© 2013 Hedge Fund Research, Inc. All rights reserved.
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References
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Brännäs, Kurt and Niklas Nordman. 2003. “Conditional Skewness Modelling for Stock Returns.”
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Campbell, John Y., Andrew W. Lo and Archie Craig MacKinlay. 1997. The Econometrics of
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Grigoletto, Matteo and Francesco Lisi. 2007. “Value-at-Risk Prediction by Higher Moment
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Hamilton, James D. 1994. Time Series Analysis. Princeton, NJ: Princeton University Press.
Heinrich, Joel. 2004. “A Guide to the Pearson Type IV Distribution.” University of
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Nagahara, Yuichi. 1999. “The PDF and CF of Pearson Type IV Distributions and the ML
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264.
Pearson, Karl. 1895. “Contributions to the Mathematical Theory of Evolution II: Skew Variation
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Stavroyiannis, Stavros, Ilias A. Makris, Vasilis N. Nikolaidis and Leonidas Zarangas. 2012.
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Encyclopedia. Accessed May 31, 2013. http://en.wikipedia.org/wiki/List_of_largest_daily_changes_in_the_S%26P_500.
Yan, Jun. 2005. “Asymmetry, Fat-tail, and Autoregressive Conditional Density in Financial
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